@Hoplite
This is a topic which I have studied pretty intently. This is a complicated discussion depending on what exactly you are asking. I will offer a brief summary below on the basis that you are just asking about how it played out the way in which it did on the 13th.
There is more than enough evidence to indicate that the attacks in the south were intended to be the main assault, but over time I have become skeptical as to whether I would call the concept of the Marye's Heights attack
completely diversionary. It wasn't intended to be the main attack, for sure, but I do think that Burnside anticipated that they would be able to seize the crossroads. The orders which issued do seem to indicate that it was believed that the crossroads could and would be taken, which would allow for more freedom of maneuver. After all, it was only being defended (initially) by a single Confederate brigade and lay only about nine hundred yards beyond the edge of the town in an advanced position from the remainder of Longstreet's Corps. It was an important point though because of the way in which the terrain and the proximity of the stone wall covering the crossroads to the edge of the town so severely constricted movement. Simultaneously, though, to be sure, the attacks would also draw attention away from what was intended to be the main attack in the south.
Again, though, that discussion aside, there is more than enough evidence to indicate that the attacks in the south were intended to be the main attacks. In fact, Franklin admitted in front of the Joint Committee on the Conduct of the War that such is what was discussed in the meeting with Burnside at Franklin's Headquarters the previous night on the 12th, but that he would need additional force to keep himself connected with the rest of the Army if he was to launch a major attack, which Burnside subsequently did by ordering two divisions from Third Corps to fulfill that responsibility.
However, when James Hardie of Burnside's staff delivered the actual order the next morning, around 7:30 A.M., or so, it appears to have left Franklin very confused as to what was intended and not without reason. We know from the testimony in front of the JCCW, that the maps which Burnside was using had mislabeled the roads.
Basically, to paraphrase, the order which Burnside wrote said that Franklin was to push forward a division, at least, being careful to keep its lines of retreat open, e.t.c. to seize Prospect Hill as a preliminary to a movement with your whole command down the Old Telegraph Road. The map which Burnside was using at Headquarters had the Old Telegraph Road running essentially to Hamilton's crossing, making the movement of a division, at least, to seize Prospect Hill a logical precursor to the rest of the attack. They would essentially seize Prospect Hill as a blocking force, while the main element of the command moved down the road towards Hamilton's Crossing, in effect hitting the flank of the Confederate Army. Unfortunately, this was not an accurate reflection of the roads. The Old Telegraph Road was not the same as the road which ran to Hamilton's Crossing. The Old Telegraph Road, as it existed in fact and as it was on Franklin's maps, ran much closer to parallel to the river, more north to south. It was actually a different road which branched off of it towards Hamilton's Crossing.
As a result, Franklin received the order and was hopelessly confused. Remember, the order stated that he was to send forward a division, at least, to "seize" Prospect Hill. Not to "carry" Prospect Hill. This implied that they were only intended to bite and hold, which is correct because the idea had been that the movement down the Old Telegraph Road was to be the main attack and the division, at least, which was to seize Prospect Hill, would essentially be covering the main attack which would be moving behind it and just south of it towards Hamilton's Crossing.
However, as noted, the Old Telegraph Road didn't actually run to Hamilton's Crossing. It ran much closer to parallel to the river, which induced Franklin to think that "the division, at least" ordered to "seize" Prospect Hill
was the attack which he was supposed to launch in preparation for a movement down the Old Telegraph Road, essentially parallel to the river. To the JCCW he basically said that from the order which he received, he could only think that Burnside had changed his mind since their meeting the previous night for his Grand Division to be responsible for launching the main attack.
Yet, they had also discussed between them that the attack was to go forward basically at dawn. Both Burnside and Franklin agree on this. However, as mentioned, for a variety of reasons, the actual order didn't actually get to Franklin's Headquarters until sometime around 7:30. The Telegraph doesn't appear to have been fully operational yet, although Telegraphic communications between Burnside's and Franklin's Headquarters would go through that day (the Telegraph was also often very spotty on the battlefield) hence Hardie having to deliver a physical copy in person, so it would have taken time for a courier to have been sent back in order to attain sufficient clarification, e.t.c.
So, at least initially, Franklin ordered a single division, Meade's Pennsylvania Reserve Division, to launch the attack to seize Prospect Hill, by itself, virtually unsupported, while the remainder of the Grand Division basically sat there prepared to move down the Old Telegraph Road, which it couldn't do without completely disconnecting from the remainder of the Army, which Franklin knew wasn't intended because Burnside had specifically acquiesced in allocating two divisions of Third Corps to ensure that a connection was maintained between Franklin's Left Grand Division and Hooker's Center Grand Division. Those two divisions would not be enough to maintain that connection if Franklin moved his Grand Division essentially southwards down the Old Telegraph Road. So, as stated, he basically just threw out a force on his left to cover his flank and kept the remainder of the Grand Division basically stationary as Meade's Pennsylvania Reserves conducted the attack. On top of this, the attack would be delayed because it became clear that the Confederate artillery prescribes was so strong that the attack was bound to fail, so they spent hours dueling with the Confederate artillery in order to try to soften up the resistance to the attack before it finally went forward. Later on, Meade's division would be supplemented by Gibbons' division, but the basic reality remained that the vast majority of the Grand Division essentially just sat back over the course of the day.
So, yeah….
Anyways, in the northern sector, Darius Couch, commanding Second Corps was assigned, via Edwin Sumner, commanding the Right Grand Division, the responsibility to conduct the attack. William French's division was assigned the responsibility to lead the attack, to be supported by W.S. Hancock's division. Due to the factors we discussed earlier, the attacks were conducted essentially a single brigade at a time, sometimes even a portion of a brigade at a time. After French's attack had failed, Hancock's division attacked behind them, but still failed to take the position. However, it was believed that the attacks in the south were ongoing, so Howard's division was then ordered thrown in. After they failed to take the position, Sturgis' division of Third Corps was ordered into the attack and failed to take the position (the Confederates were actively reinforcing it). Burnside then learned that the attacks in the south had failed, to which he ordered that it be recommenced and Burnside ordered Griffin's division into the attack. Franklin basically responded saying that he could not progress with another attack, as two divisions had already been used up, e.t.c. By the time that Burnside had received this, Griffin's division was already attacking. Burnside subsequently ordered Franklin again to renew the attack and consequently sent Humphrey's division in. They would then be followed at twilight by Hawkin's brigade of Getty's division. After the sun had set, Burnside learned that Franklin had again declined to renew the attacks in the south.
With the assessment in hand from Franklin that the attacks could not be reasonably renewed in the south, Burnside considered shifting the emphasis of the attack to the northern sector for the 14th. Multiple accounts stated that he even considered personally leading his old Ninth Corps in the attack. However, he didn't want to continue with the piecemeal attacks, which, due to the terrain and proximity of the edge of the town to the Stone Wall, would have meant a long and arduous process in trying to get the command into positions from which they could launch a more concerted attack. Some of the command would probably have to do so under fire, as well. Eventually, under the advice of his subordinates, Burnside cancelled the planned attacks for the 14th and elected to fall back across the Rappahannock that night (the night of the 14th-15th), which they did.
That doesn't get into all of the fighting. There was some nasty fighting along Deep Run in the afternoon, following the repulse of Gibbon's division, involving elements of Hood's division, particularly several regiments of Evander Law's brigade, but that does provide a basic synopsis that way.
So, hopefully that is helpful.